Don't know if this will help answer your question, but here you go:The purpose of preburners and heat exchangers in the context of a rocket engine is to add more energy to the fuel flow, almost always to drive a turbine which in turns takes some of that energy and uses it to drive a pump to move more fuel through the system. That's what makes it a "cycle". In an expander cycle engine, energy is added to the fuel via the cooling channels surrounding the nozzle/combustion chamber. The nozzle is acting as a heat exchanger, exchanging heat from the rocket exhaust into the fuel flow; in other words, adding energy to the fuel while taking it away from the exhaust. The thermal energy of the exhaust is transferred into the fuel flow, and then converted into mechanical energy via a turbine. This is a "two birds with one stone" solution since it adds energy to our fuel flow, allowing us to drive our pump, while also keeping the nozzle cool. The expense of an expander cycle engine is that you are taking some of the thermal energy out of the exhaust.In a staged-combustion cycle engine, a small portion of the fuel and oxidizer are burned in order to release their chemical energy and drive a turbine. This turbine takes that released chemical energy and turns it into mechanical energy which drives a pump to pump more fuel/oxidizer into the engine. There's your cycle again. We can also take advantage of regenerative cooling, just like in an expander cycle engine because we don't want to melt our nozzle. The cost of the staged combustion cycle is that we are taking some of our fuel that could be burned in the combustion chamber, and instead using it to drive a turbine to pump more fuel into the system.Now, with a full flow expander cycle engine, you run into the challenge that all of your "pumping energy" needs to come from your heat exchanger. When trying to get enough energy to meaningfully drive fuel/oxidizer, your heat exchanger is now the bottleneck, and you aren't able to harvest enough energy due to limitations in the size/geometry of the engine. You'll end up with a "chicken and the egg" scenario where to drive the pumps, you need more energy from the combustion chamber, but to get more combustion energy, you need to pump more.With staged combustion, you can always burn more fuel/oxidizer if you need more pumping power since the "pumping energy" is generated independent from the main combustion process of the engine.
And to complicate things advanced expander cycle engines that have recently been researched and developed as listed in NTRS etal can expand capabilities by adding one or more additional heat exchangers into the circuit.
Oh, ok. I think I get it now?
The raptor is technically dual staged full flow with single-sided (CH4) expander cycle because LCH4 cooling is needed for the throat, chamber, and nozzle. That energy doesn't go to waste, I suspect the CH4 pump uses slightly less power than the simple calculation case because of the heat transferred to the coolant loop.But flowing oxygen to cool things? Yikes, the materials for just the oxygen preburner and pumps are already exotic enough, you have to severely limit the surface area of that to keep reliability and costs to a reasonable level. You'll note on the Raptor diagram the smallest surface area is the LOX preburner/pump side. It's straight in to the combustion chamber ASAP.Oxygen really wants to oxidize, that's what it does, and it'll do it to your metal cooling channels quite happily.
So LOX cooling is almost as efficient as a preburner?
Oh. It’s just simpler and more appealing to work with than a preburner because it’s easier to handle?